Introduction

Flux pinning has been a topic of much interest in the field of superconductivity because of its importance for applications and aspects of fundamental physics. This interest stems from the significance of flux pinning for high critical current density (Jc) in superconductors, which is the defining property of a superconductor. Generally, various types of random imperfections, such as cold-work-induced dislocations, secondary-phase precipitates, defects induced by high energy ion irradiation, etc., can be used to enhance flux pinning. Unfortunately, it is difficult to discern the maximum potential of a superconductor from these techniques, and the outcomes hold up only to a certain level. Furthermore, the critical current is only enhanced, in most cases, either in low or high fields, but not in both, while degradation of the superconducting critical temperature (Tc) is another drawback. For instance, proton irradiation can only enhance flux pinning in high fields by inducing point defects in K:Ba122 1. Similarly, light ion C4+ irradiation of Ba122:Ni crystals can only enhance Jc in low fields at high temperatures2. High energy particle irradiation can also decrease the critical superconducting temperature (Tc) by more than 5 K for cobalt and nickel doped Ba-122 3,4.

As is well known, Jc is mostly limited by weak links (in the case of polycrystalline bulks), and thermally activated flux creep (an intrinsic property) emerges from weak pinning5,6,7,8,9,10,11. Strong pinning can be achieved by inducing effective pinning centres with strong pinning force. Our previous results show that Jc is enhanced significantly under hydrostatic pressure in high fields (i.e., over one order of magnitude) in comparison to low fields, along with enhancement of the closely related Tc by more than 5 K in Sr4V2O6Fe2As2 polycrystalline bulks and NaFe0.97Co0.03As single crystals12,13. Until now, however, it has been unclear whether the observed Jc enhancement under pressure is correlated with improved Tc or flux pinning. The primary motivation for the present work is to use optimally doped single crystal samples (which have an unchanged Tc under hydrostatic pressure) to elucidate the contributions of flux pinning to Jc enhancement in Fe-based superconductors. The secondary motivation is to investigate further the contributions from both the pinning centre number density (Np) and the pinning force (Fp) to strong pinning.

The argument is as follows: Hydrostatic pressure can induce pinning centres, which, in turn, enhance the pinning force. The total pinning force and the pinning centres are correlated by Fp = Npfp where Np is the number density of pinning centres and fp is the elementary pinning force, defined as the maximum pinning strength of an individual pinning centre, with a value that depends on the interaction of the flux line with the defect. According to the flux pinning theory, strongly interacting defects can contribute to Fp individually, provided that Fp ∝ Np, and weakly interacting defects can contribute only collectively; the collective theory therefore gives Fp ∝ (Np)2 for small defect numbers14.

K:Ba122 compound is believed to be the most technologically suitable because of its isotropic nature and high Tc, upper critical field (Hc2), and Jc values (Jc > 106 A/cm2 at 2 K and 0 T)15,16,17,18,19. According to the Ginzburg-Landau theory, the depairing current density (Jd) is the maximum current density that superconducting electrons can support before de-pairing of Cooper pairs, and is given as

where Φo is the flux quantum and μo is the permeability constant. The Jd value that is found is roughly 0.3 GA/cm2 by using the following values of the penetration depth, λ = 105 nm and the coherence length, ξ = 2.7 nm20,21. Our estimation indicates that there is a considerable potential to further enhance flux pinning in (Ba,K)Fe2As2.

In this paper, we investigate the flux pinning of optimally doped (Ba,K)Fe2As2 under hydrostatic pressure. We demonstrate that hydrostatic pressure causes little change in Tc, but leads to significant enhancement in flux pinning or Jc by a factor of 5 in both low and high fields in optimally doped Ba0.6K0.4Fe2As2 crystals. At 4.1 K, high pressure can significantly enhance Jc from 5 × 105A/cm2 to nearly 106A/cm2 at 2 T and from 2 × 105A/cm2 to nearly 5.5 × 105A/cm2 at 12 T. Our systematic analysis shows that the both Np and Fp are increased by the pressure and contribute to strong pinning.

Figure 1 shows the temperature dependence of the magnetic moments for zero-field-cooled (ZFC) and field-cooled (FC) measurements at different pressures. Tc remains almost unchanged at different pressures. Tc ≈ 37.95 K was found at P = 0 GPa and P = 1 GPa. Similar results were also reported for Ba0.6K0.4Fe2As2 thin film22. Furthermore, a temperature independent magnetic moment at low temperatures was observed, along-with a small transition width, indicating the high quality of the crystals.

Figure 1: Magnetic moments vs. temperature at P = 0 GPa and P = 1.2 GPa.
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Tc remains almost unchanged at different pressures. Tc ≈ 37.95 K was found at P = 0 GPa and P = 1 GPa.

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The field dependence of Jc at different temperatures (4.1, 16, and 24 K) and pressures (0 and 1.2 GPa), obtained from the magnetic hysteresis (M-H) curves by using Bean’s model, are shown in Fig. 2. Nearly five-fold Jc enhancement can be seen at 16 K and 24 K in both low and high fields at P = 1.2 GPa. It is noteworthy that Jc is enhanced for the Ba0.6K0.4Fe2As2 crystal at 1.2 GPa in both low and high fields. This has not been found with the other approaches for pinning enhancement reported so far. At 16 K and self-field, the Jc is 2 × 105A/cm2 and it increases up to 6 × 105A/cm2 under pressure of 1.2 Gpa, with as high a value as 3 × 105A/cm2 retained at 12 T. At 24 K, Jc at zero field is 9 × 104A/cm2 which increases to 2.5 × 105A/cm2 at P = 1.2 Gpa, with the same value retained at 12 T. At 4.1 K, the Jc is nearly 1 × 106A/cm2 at 2 T and 5 × 105A/cm2 at 12 T under P = 1.2 GPa.

Figure 2: Jc as a function of field at P = 0 and 1.2 GPa at 4.1, 16, and 24 K.
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Jc is improved in both low and high fields and nearly five-fold Jc enhancement can be seen at 16 K and 24 K in both low and high fields at P = 1.2GPa.

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The pinning force (Fp = Jc × B) as a function of field at 8 K, 12 K, 24 K, and 28 K is shown in Fig. 3 23. At high fields and pressures, the Fp is found to be nearly 5 times higher at 8, 12, 24, and 28 K as compared to the corresponding value at P = 0 GPa, which agrees nicely with the Jc enhancement results. Figure 4 shows a comparison of Fp obtained in our Ba0.6K0.4Fe2As2 under pressure with those of several other low and high temperature superconducting materials24,25,26,27. The (Ba,K)Fe2As2 shows better in-field performance under pressure. Pressure can significantly improve Fp values to greater than 60 GN/m3 at H > 10 T, which are even superior to those of Nb3Sn and NbTi.

Figure 3: Fp versus field at 8, 12, 24, and 28 K at different pressures.
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At high fields and pressures, the Fp is found to be nearly 5 times higher at 8, 12, 24, and 28 K as compared to the corresponding value at P = 0 GPa, which agrees nicely with the Jc enhancement results.

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Figure 4: Comparison of Fp for different superconductors.
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Pressure can significantly improve Fp values to greater than 60 GN/m3 at H > 10 T, which are even superior to those of Nb3Sn and NbTi.

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With respect to the Np, pressure can also increase the number of point pinning centres (point defects), which can suppress thermally activated flux creep, leading to Jc enhancement12. Np is calculated by using the following equation28:

where ∑Fp is the accumulated pinning force density, is the maximum elementary pinning force (fp), which is the interaction between a flux line and a single defect, and η is an efficiency factor. η = 1 corresponds to a plastic lattice, and the η value is otherwise where B is the bulk modulus of the sample. We assume to a second order of approximation that the interaction between a flux line and a single defect is nearly the same under pressure. Therefore, we can use ≈ 3 × 10−13N for a similar superconductor (i.e., Ba122:Co) to estimate Np 29. At 4.1 K, Np ≈ 7.3 × 1024m−3 at P = 0 Gpa, which increases to Np ≈ 1.2 × 1025m−3 for P = 1.2 GPa, while at 24 K, Np ≈ 6.6 × 1023m−3 at P = 0 Gpa, which increases to Np ≈ 3.8 × 1024/m3 for P = 1.2 GPa.

In order to examine if the pinning force enhancement is the major factor responsible for the observed Jc enhancement in our crystal under pressure, we have calculated the differences in the ratios of and and plot the results in Fig. 5 as a function of field. Analysis of the data, acquired at different temperatures, leads to values of nearly zero. This result indicates that Jc enhancement is only related to pinning force enhancement.

Figure 5: Jc−Fp ratios at P = 1.2 GPa and P = 0 GPa.
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The relative change of lnJc with pressure as a function of T is given in the inset. Analysis of the Jc-Fp ratios, acquired at different temperatures, leads to values of nearly zero. This result indicates that Jc enhancement is only related to pinning force enhancement.

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To examine whether the observed Jc enhancement is likely to be affected by volume change of the samples under high pressure, we have performed the following analysis. According to the Wentzel-Kramers-Brillouin (WKB) approximation, high pressure can affect the grain boundaries by reducing the tunnelling barrier width (W) and the tunnelling barrier height (L) for polycrystalline bulks, in accordance with the following simple mathematical expression30,31,32:

here k = (2 mL)1/2/ corresponds to the decay constant, where is the reduced Planck constant, and Jc0 is the critical current density at 0 K and 0 T. The relative pressure dependence of Jc can be determined from Eq. (3) as33:

The reduction in the width and height of the grain boundaries can be written as and , respectively.

We can use this model for the (Ba,K)Fe2As2 single crystals, by assuming to a first approximation that κGB and κL can be nearly equated to the average linear compressibility values κa = −dlna/dP (κa ≈ 0.00318 GPa−1) and κc = −dlnc/dP (κc ≈ 0.00622 GPa−1), respectively, in the FeAs plane, where a and c are the in-plane and out-of-plane lattice parameters34. Consequently, Eq. (4) can be modified as

By using Jc ≈ 105A/cm2 at 24 K and Jc0 ≈ 106A/cm2, ≈ 0.0073 GPa−1 and (1/2 ) ≈ 0.0071 GPa−1, which contribute collectively not more than 2% of the experimentally obtained value, i.e., dlnJc/dP = 0.92 GPa−1 from the inset of Fig. 5. This illustrates that the source of the flux pinning under pressure is not the volume change.

The Jc value vs. reduced temperature (i.e. 1-T/Tc) at 0 and 10 T under different pressures is shown in Fig. 6. The data points in different fields and pressures follow a power law description [i.e. Jc ∝ (1 − T/Tc)β], where β is a critical exponent35,36,37. At specific fields, Ginzburg-Landau theory predicts distinct vortex pinning mechanisms, with different values of exponent β. For example β = 1 corresponds to non-interacting vortices and β ≥ 1.5 corresponds to the core pinning mechanism. Our value of β ~ 1.74 and 1.85 for zero field, and β ~ 1.20 and 1.43 at 10 T, at 0 and 1.2 GPa, respectively, reveal a robust dependence of Jc on pressure. The low β values at high pressure show the weak field dependences of Jc in contrast to its values at low pressure. Different values of exponent β have also been observed in MgB2 and yttrium barium copper oxide (YBCO)38,39.

Figure 6: lnJc versus reduced temperature at different fields and pressures.
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The low β values at high pressure show the weak field dependences of Jc in contrast to its values at low pressure.

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The pinning mechanisms in Ba0.6K0.4Fe2As2 have been examined in the frame of collective pinning theory. Generally, core pinning comprises 1) δl pinning, which comes from spatial variation in the charge carrier mean free path, l, and 2) δTc pinning due to randomly distributed spatial variation in Tc.

Referring to the Griessen et al. approach:

corresponds to pinning, while

applies in the case of δTc pinning, where t = T/Tc 40. Figure 7 shows almost perfect overlapping of the experimentally obtained Jc values and the theoretically expected variation in the δl pinning mechanism at 0.05 T. This is in agreement with the observation of little change in Tc under high pressure. We also observed similar results in BaFe1.9Ni0.1As2 and SiCl4 doped MgB2 41,42. Furthermore, δl pinning has also been reported in FeTe0.7Se0.3 crystals43.

Figure 7: Jc as a function of T/Tc.
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Experimental data points are in good agreement with δl pinning.

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In conclusion, we have systematically examined the flux pinning in optimally doped Ba0.6K0.4Fe2As2 crystal under hydrostatic pressure, analyzing the critical current density that was determined experimentally. We have demonstrated that strong flux pinning in both low and high fields can be achieved by improving the pinning force under pressure. The pressure of 1.2 GPa improved the Fp by nearly 5 times at 8, 12, 24, and 28 K, which can increase Jc by nearly two-fold at 4.1 K and five-fold at 16 K and 24 K over a wide range of fields. This study also demonstrates that the performance of an optimally doped superconductor in both low and high fields can also be further enhanced by pressure.

Experimental

High quality 122 crystals were grown by the flux method. The pure elements Ba, K, Fe, As, and Sn were mixed in a mol ratio of Ba1−xKxFe2As2:Sn = 1:45–50. A crucible with a lid was used to control the evaporation loss of potassium along with that of arsenic during growth. The crucible was sealed in a quartz ampoule filled with Ar and loaded into a box furnace15. The M-H loops at different temperatures and pressures and the temperature dependence of the magnetic moments were measured on a Quantum Design Physical Properties Measurement System (QD PPMS 14 T) by using the Vibrating Sample Magnetometer (VSM) option. We used an HMD high pressure cell and Daphne 7373 oil as the medium for applying hydrostatic pressure on our samples. Further details can be found in pressure cell manual i.e. Quantum Design (QD) High Pressure Cell User Manual for use with the QD VSM, No. CC-Spr-Φ8.5D-MC4. The magnetic fields were applied parallel (H//ab) to the ab-plane of the samples.

Additional Information

How to cite this article: Shabbir, B. et al. Study of flux pinning mechanism under hydrostatic pressure in optimally doped (Ba,K)Fe2As2 single crystals. Sci. Rep. 6, 23044; doi: 10.1038/srep23044 (2016).